How to evaluate:

∫∫∫ xyz/(x^2+y^2)^1/2 dv

E

where E is the solid "quarter-cylinder" in the first octant bounded by a cylinder of radius a>o with axis long the z axis, the planes y=0,x=0,z=0 and z=b, wiht b>0.

Thankyou very much!

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- Nov 20th 2008, 09:59 AMiwondercylindrical or spherical coordinates problem
How to evaluate:

∫∫∫ xyz/(x^2+y^2)^1/2 dv

E

where E is the solid "quarter-cylinder" in the first octant bounded by a cylinder of radius a>o with axis long the z axis, the planes y=0,x=0,z=0 and z=b, wiht b>0.

Thankyou very much! - Nov 20th 2008, 11:23 AMJD-Styles
The best way to tackle this one is to change to cylindrical coordinates. So we'd get:

$\displaystyle

\int_0^b \int_0^{\pi /2} \int_0^a \frac{(\rho cos\phi )(\rho sin\phi ) z}{\rho } \rho d\rho d\phi dz

$

$\displaystyle

\int_0^b zdz \int_0^{\pi /2} cos\phi sin\phi d\phi \int_0^a \rho ^2 d\rho

$

$\displaystyle

(\frac{b^2}{2})(\frac{1}{2})(\frac{a^3}{3})

$

$\displaystyle

\frac{a^3 b^2}{12}

$ - Nov 20th 2008, 09:58 PMiwonder
thankyou!

i got it so far, but how do u get p^2 for the third limit of integration? is that only be p??? - Nov 23rd 2008, 09:22 AMJD-Styles
There's one p from x, on from y, one from the change to spherical coordinates (that comes with d phi), so that's p^3 in the numerator. Then in the denominator you have one p from (x^2+y^2)^1/2. So in total you have p^(3-1)=p^2